Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj96 Structured version   Unicode version

Theorem bnj96 29236
Description: Technical lemma for bnj150 29247. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj96.1  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj96  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Distinct variable groups:    x, A    x, R
Allowed substitution hint:    F( x)

Proof of Theorem bnj96
StepHypRef Expression
1 bnj93 29234 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 dmsnopg 5341 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
31, 2syl 16 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
4 bnj96.1 . . 3  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54dmeqi 5071 . 2  |-  dom  F  =  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }
6 df1o2 6736 . 2  |-  1o  =  { (/) }
73, 5, 63eqtr4g 2493 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   {csn 3814   <.cop 3817   dom cdm 4878   1oc1o 6717    predc-bnj14 29052    FrSe w-bnj15 29056
This theorem is referenced by:  bnj150  29247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-suc 4587  df-dm 4888  df-1o 6724  df-bnj13 29055  df-bnj15 29057
  Copyright terms: Public domain W3C validator